Saturday, June 29, 2013

this morning we were talking about work. she happened to have a few blueprints(?) in her bag of what her firm was doing and how many simultaneous processes the proposed factory would be using.

it seemed very complicated to me .. like an electric circuit diagram on steroids!

then an idea came to mind .. maybe it can be my turn! .. so on her computer i showed her one of my preprints. her brow furrowed.

the basic idea's isn't that bad, i tell her.

for things that behave like mass distributions, the small-scale behavior can sometimes determine the large-scale. if, for example, there aren't enough directions when you zoom in at any given point, then the mass has to be distributed either like dust [1] or be restricted to fewer dimensions, like a line in space.

we then talked about how, at a certain point, maths no longer consists of calculating things .. not exactly, anyway.

[1]
i.e. a fractal with non-integer hausdοrff dimensiοn, but not necessarily self-simιlar. at the time i was thinking about purely unrectifιable subsets of the plane.

[2]
thinking about it, maybe i should have said something like: well, define "dust" ..?

*NOTE* the new label ½ × 2 is meant to tag those (few) posts which are about how mathematics, for inexplicable reasons, relates to my relationships. the idea is: once you've been with someone long enough, you are no longer 1 person. rather, you are one-half of two people. (-:

Monday, June 24, 2013

well, it's been 25 of these roundups. my guess, as before, is that you readers can probably do your own news searches, and with greater effectiveness to fit your own tastes.

that said, i'll still be posting articles i find interesting, but just not in roundup form anymore.

1. not the isomorphism that i expected, but ..

.. apparently some processes are universal, whether organic or digital:

"On the surface, ants and the Internet don't seem to have much in common. But two Stanford researchers have discovered that a species of harvester ants determine how many foragers to send out of the nest in much the same way that Internet protocols discover how much bandwidth is available for the transfer of data."

why is this even a question? we haven't stopped teaching spelling, grammar, and vocabulary to students, just because they now have word processors, right?

The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth. The idea is that competence with algorithms can be substituted for by the use of calculators, and reformists often call for training students in the use of calculators as early as first or second grade.
..
That the use of standard algorithms isn’t merely mechanical is not by itself a reason to teach them. It is important to teach them because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts.

admittedly, though, a lot of times i just opine when i feel like it, and drop things when i don't ..

These professors maintain that college-level work requires ready and effortless competence with the standard algorithms and that the student who needs to ponder fractions — or is dependent on a calculator — is simply not prepared for college math. They express outrage and bafflement that so much American math education policy is set by people with no special knowledge of the discipline.

to be honest, i don't know who really is in charge of mathematics educational policy, but i strongly suspect that most college faculty opt out of any involvement with it. call it a professional opportunity cost: unless you rake in a lot of external funding from it, there's not a lot of incentive to work on educational problems instead of research ones.

3. wait .. what?

there are plenty of shocking items of news in the world lately, like:

This year, a pilot scheme was introduced to enforce the rules.
..
When students at the No.3 high school in Zhongxiang arrived to sit their exams this month, they were dismayed to find that they would be supervised by 54 randomly selected external invigilators.
..
By late afternoon, the invigilators were trapped as students pelted the windows with rocks. Outside, more than 2000 people had gathered, smashing cars and chanting: ''We want fairness. There is no fairness if you do not let us cheat.''
..
The protesters claim cheating is endemic in China and that sitting the exams without help puts their children at a disadvantage.

the last excerpt warrants interpretation; i think the protest is specific to why their city was chosen for enforcement, instead of other cities. the article goes on to point out that last year, the education department received 99 identical exam papers .. but then again, there's no comparison as to how other cities' stats stack up.

imagine, for example, if other cities had been recorded at 500+ identical copies.

considering the population of certain cities in china, that's no longer a large number of students;
as a result, without additional data i wouldn't rule out political favoritism yet ..

nonetheless, this kind of news is alarming. i've heard that eastern culture favors the group over the individual, and that the socioeconomic inequity in india and china is huge, but .. really?

well, moving towards a more abstract direction ..

The researchers are trying to avoid a situation where we outsmart ourselves, and create a system that can in turn invent its own technologies, which could "steamroll" humanity – not because it’s evil, but simply because we couldn’t foresee the long-term ramifications of how we programmed it.
..
"Think how it might be to compete for resources with the dominant species," Price says. "Take gorillas, for example – the reason they’re becoming extinct isn’t because humans are actively hostile towards them, but because we control the environment in a way that suits us, but is detrimental to their survival.

considering how well we understand turιng machines yet still face computer crashes regularly, in this day and age, this might not be a bad approach in general. it would be rather .. depressing if all of humanity were swept away, due to a computer glitch!

to a lesser, more realistic extent: the access and operations for my savings and checking accounts are probably automated within some banking computer system. if enough errors pile up, then .. [cringes].

4. to humans, does time lack symmetry?

i would have thought that we humans are good at accounting for symmetry. then again, i'd probably also fall prey to this ..

But when asked to predict what their personalities and tastes would be like in 10 years, people of all ages consistently played down the potential changes ahead.
..
Thus, the typical 20-year-old woman’s predictions for her next decade were not nearly as radical as the typical 30-year-old woman’s recollection of how much she had changed in her 20s. This sort of discrepancy persisted among respondents all the way into their 60s.

the main mechanism might be memory, which resides clearly in the past, not the present, and is arguably inherently faulty.

anyway, towards a more familiar setting .. here's a thought that makes sense, but never came to mind when i was teaching:

You would think that since you have been a student and survived you would be able to recognize their misconceptions and guide them to enlightenment. But even if you could remember what it was like to be a student, that moment for you was characterized by a similar hit-and-run-don’t-leave-a-calling-card confusion.

Everybody gets hit by the bus, everybody gets knocked out, everybody survives, but nobody remembers what they got hit by. So you can’t tell them what to watch out for. Teachers can’t understand their students’ confusion even though they once experienced the exact same confusion!

a good point, but the comparison (i.e. hit-&-run) as well as the phenomenon seems a bit forced. it's not like the act of learning is some version of achilles and the tortoise, right!

maybe the point is the instability of learning. everything is fine for students, as long as they're following diligently and carefully .. but once they trip up, their confusion can be utter confusion. /-:

5. lastly, a bit about online education.

i know that i've ranted on and on about MOOCs a lot, but this is the first article that mentions any behind-the-scenes kind of details:

So, what hasn’t gone as planned? Certainly some things do not translate from a traditional classroom course to a MOOC. Our team realized quickly that we needed to do a better job cross-linking material on the course site. For example, if we mention the syllabus, we must link to it. Some students, we have learned, want a great deal of guidance.
..
We also underestimated the misunderstandings that can arise from idiomatic and discipline-specific language. We began the course by asking students to complete a Personal Benchmark Statement, only to discover that we needed to provide a definition of “benchmark.” A longer glossary of terms became a featured part of our site.

interesting: in terms of the latter gaffe, in a multivariable calculus class it once took me a week to realise that i'd been calling a $\partial$ a 'del' without actually having told the class of the pronunciation.

(for the record, it was one very brave student who asked in the middle of class; subsequently a collective 'oh' erupted through the lecture hall.)

Sunday, June 23, 2013

all i did today was make coffee, read a little, and sort out photos from yesterday. i didn't do any maths.

today is midsummer's day (or juhannus) and yesterday was midsummer's eve. both are official holidays in finland and arguably the most important ones. celebrating it has the same, easy feeling as thanksgiving in the u.s.: you don't have to believe in anything, other than the fact that the longest day of the year is upon us, and one observes it during the closest weekend to it.

i used to work through holidays or, at the very least, try to do so as much as possible .. where "possible" depended on whose company i kept. living in finland these last two years, though, has tempered that desperate feeling.

i'd like to think that i'm a tireless, relentless researcher, but that's hardly true. for instance, i found out that there's only so much time in a single day that i can spend thinking, reading, attacking a certain line of thought.

the truly productive hours are preciously few, in fact: usually two hours in the morning and two in the late afternoon or early evening. maybe the former is fueled by an early surge of coffee and dopamine, the latter by the mere suggestion that the end of the day's work is near and sh-t, i'd better get something done![1].

it is hard to ignore the regular workday, even if one isn't beholden to it. during the academic year the department is often quite empty by 4pm, if only because that's when kids get off from school and the faculty, most of whom are parents, are off to pick them up!

in contrast, i remember teaching 2+2 during my first postdoc, going home exhausted, and just being too mentally drained to get any real thinking done. [2] i was always struggling to find long blocks of time in order to get something done, only to regularly come up with nothing of consequence.

it got to the point where i really wondered if i was really right for this job, that maybe i wasn't meant to be a mathematician. heck, i still don't know if i'm right for this job. i mean, it's working well enough .. but often i fear for the next dry spell of inspiration. even during holidays now, it's hard to let go completely and just forget.

at any rate, i now know that there's an opposite to the spectrum: sometimes i feel like i have too much time and it's wasted: i can neither use it for research gain, and it's probably not enough to warrant taking on a partial load of teaching duties ..

[1]
there are exceptions, of course. if i'm in the middle of a draft of a paper, then the hours fly by and i have to take care, set the work aside, and do something else before going to bed at night .. if only to guarantee that i'll sleep well enough to work efficiently the next day.

there's a difference, i suppose, between the initial understanding and development of the problem and, upon success, the technical implementation of its solution.

[2]
it took me a while to get used to the routine of lecturing and developing plans for enough of the standard courses, before i could "shut off" that part of my brain during my non-teaching hours. sometimes i suspect that i'm just highly inefficient at scheduling and multi-tasking.)

now this is the kind of mathematical biology that i like to see .. the kind with geοmetric measνre theοry in it!

// as indicated by the link below, this preprint is a few weeks oldl
i stumbled upon it on 5 june 2013.

Beside the obvious geοmetric intrinsic interest such a minimization under isοperimetric and geηus constraint could have, a motivation to study this problem comes from the mοdelization of the free energy of elastic lipid bilayer membranes in cell biοlogy.Indeed the Willmοre functiοnal is closely related to the Helfrιch functional which describes the free energy of a closed lipid bιlayer
$$
F_\text{Helfrich} \;=\;
\int_\text{lipid bilayer}
\left(
\frac{k_c}{2}(2H+c_0)^2 + \bar{k}K+ \lambda
\right) + p \cdot V
$$
where $k_c$ and $\bar{k}$ denote bending rιgidities, $c_0$ stands for the spontaneous curνature, $\lambda$ is the surface tensiοn, $K$ and $H$ denote as usual the Gauss curνature and the mean curνature, respectively, $p$ denotes the οsmotic pressure and $V$ denotes the enclosed volume. The shapes of such membranes at equilibrium are then given by the corresponding Euler-Lagraηge equation. If $c_0 = \lambda = p = 0$ the Willmοre functiοnal captures the leading terms in Helfrich's functional (up to a topolοgical constant). Whereas if these physical constants do not vanish, $\lambda$ and $p$ can be seen as Lagrange multipliers for area and volume constraints. Thus, thanks to the invariance under rescaling of both the Willmοre functiοnal and the isοperimetric ratio, we exactly face the problem of minimizing the Willmοre functiοnal under an isοperimetric constraint.

In the context of vesιcles, imposing a fixed area and a fixed volume has perfect biological meaning: on one hand, it is observed that at experimental time scales the lipid bilayers exchange only few molecules with the ambient and the possible contribution to the elastic energy due to displacements within the membrane is negligible. Thus, the area of the vesιcle can be treated as a fixed one.On the other hand, a change in volume would be the result of a transfer of liquid into or out of the vesicle. But this would significantly change the οsmotic pressure and thus would lead to an energy change of much bigger scale than the scale of bending energy.

At first glimpse one may think that biologically relevant vesicles should always be of spherical shape. But in fact also higher geηus membranes are observed:
for tοroidal shapes see [43] and [60], for geηus two surfaces see [37], and for higher geηuses see [38]. Further details can be found also in [34]..

Monday, June 17, 2013

i'm starting to think that these roundups are a little .. artificial; maybe 'unfulfilling' is a better word. honestly, most of these posts have nothing to do with maths and are hardly instructive.

also, sometimes it's a struggle to find enough interesting articles to share, every week. other times there are too many to choose, and to list them all would contribute little more than social media websites like /. or reddit .. and less effectively, at that.

i used to post articles that were interesting to me upon finding them, and sometimes added commentary of some depth. maybe i'll switch back to that, especially as two of the shared links below are already in that format. (if you have a constructively critical opinion on the matter, then let me know in the comments.)

at any rate, this is what i found this week:

well, it's not the fields medal ..

.. but i suppose it's nice to get recognition for your work, even if it's from your local congressperson!

"The congressman concluded by congratulating Philip T. Grεssman and Robert M. Straιn of Penn State’s Department of Mathematics for solving the equation. Another odd thing about McNerneγ’s speech on the House floor? The “new advancement in mathematics” is about three years old."

I’m not talking about the kind of travel that neatly fills those allotted twelve days of annual vacation. No, that kind of travel is frenzied, restricting the majority of life to an unnatural cycle of constant want of more. I’m talking about the kind of rugged, unplanned long-term travel where you give up owning most things, leave behind a stable home, learn to live simply on a budget, and really see the world for months on end. The kind of travel that was possible after graduation, when you strapped on your backpack and jumped into the unknown world, flowing carefree wherever the wind blew.

i've been telling my colleagues that i am looking forward to a boring summer;
this means, in particular, no unnecessary traveling for work [1].

boring is an under-rated word. it belies qualities like safe and predictable, of course, which are qualities much maligned in this modern age that favors innovation and novelty.

my opinion is different, if only because it is personal. i already strive for plenty of innovation in my work. i don't need anymore creeping into the rest of my life. every day of research is a cumulative attempt of seeking new explanations of phenomena that i just don't understand.

to give you an idea of my time frame:

every few months, i make a small discovery or two ..
.. but nothing that really clears everything up.

i've been thinking about the same fundamental problems for about .. 5-6, maybe 7 years, and i still don't have a full answer. i wish i did [2].

to put it bluntly, i spend most of my time confused at ideas.
there's no easy way out of this, either.

my field of expertise is so tiny (read: insignificant) that probably only a dozen or two people in this world are probably familiar with the same ideas. plus, they don't always reply to their emails. the down-side is that, on average, there's nobody else around that can explain things to me, which means that i have to "think my way out" by myself and grope in the proverbial dark, alone.

so though i'm not exactly a creature of habit, i do appreciate the familiarity in my daily life. traveling for work is just that much more unfamiliarity: on top of being confused at infinite-dimensional Banach spaces, the last things i want are to remember where my hotel or hostel is, what the words for coffee and lunch are in the native language, and so on.

i'm no apologist for why life is boring. if you want to live a life of adventure and innovation, then go ahead: i'll just stick to my lonely, esoteric, academic journey, thank you very much, and enjoy predictably fine espresso, at the pleasant cafes on the street corners that i know relatively well.

the stuff nightmares are made of.

i remember a few fellow ph.d. students from china and other chinese students during my first postdoc. as the summer went on, everyone else left to see their families and take vacations. they, on the other hand, might take a road trip through the u.s. or just stay around.

when i asked them why they didn't go back and visit their families, they said that there might be visa problems, coming back: they might have to wait a few months before their paperwork was in order .. or in rare cases, they wouldn't be let back at all.

i never knew how likely the latter possibility would happen .. but it's a painful thing to hear that problems can come from the u.s. side:

"Οmar F. Zaιdan, a Jordanian citizen, was denied re-entry to the US on the eve of his PhD defense at Johns Hopkins University. It has been over a year and a half, and he has not yet been allowed to return."

admittedly, i would have thought that a lack of comfort with ambiguity would actually make for better thinking, because one would be more motivated to look for the right answer.

on the other hand, if it's a problem worth solving, then it's probably hard enough that you will spend days, weeks .. even years .. working out out, which is a lot of time suffering in uncertainty.

there's also the contribution offered by creativity too, i suppose;
new ideas rarely come from overly rigid thinking.

"Afterwards, each participant filled out a survey measuring their emotional need for certainty and stability. They expressed their agreement or disagreement with such statements as “I don’t like situations that are uncertain” and “I dislike questions that can be answered in many different ways.”
..
Those who read a short story had significantly lower scores on that test than those who read an essay. Specifically, they expressed less need for order and more comfort with ambiguity. This effect was particularly pronounced among those who reported being frequent readers of either fiction or non-fiction."

somehow reading this article outraged me in a way that i cannot explain, which suggests that my reaction is purely emotional. conceptually it makes sense to me that, in a market economy, value and utility don't necessarily go hand in hand. however, i cannot help but feel a certain way about what looks fair:

i'm not one to talk, though:

i'm a university researcher whose work isn't readily applicable to society for the foreseeable future (i.e. useless). if i had to measure my utility to the economy, in the here and now, then the main contribution would actually come from my teaching instead of my research.

a lot of times i remind myself of a kind of hippocratic oath, that at least i'm not doing too much harm. i'm not directly making rich people richer, nor contributing to the further inequity suffered by disenfranchised minorities.

one could argue that i am causing harm in terms of opportunity cost: the fact that the nation of finland is paying me to work on 'useless stuff' means that less funding goes to biologists who might cure cancer, sociologists who could study how to eradicate poverty, or engineers who could find new ways to harness renewable energy or clean up the environment [1]. since i took the money, that does put a little bit of responsibility upon me to do my job well .. even if other jobs could produce more immediate societal benefit.

another passage from the article now becomes relevant:

"I do most of that work with a tool called Ruby on Rails. Ruby on Rails does for web developers what a toilet-installing robot would do for plumbers. (Web development is more like plumbing than any of us, perched in front of two slick monitors, would care to admit.) It makes tasks that used to take months take hours. And the important thing to understand is that I am merely a user of this thing. I didn’t make it. I just read the instruction manual. In fact, I’m especially coveted in the job market because I read the instruction manual particularly carefully. Because I’m assiduous and patient with instruction manuals in general. But that’s all there is to it."

the phrase "i just read the instruction manual" is especially poignant to me.

sometimes i feel the same way: if you read the same research articles and textbooks that i read, thought about the same problems for just as long as i have, then you'd probably have reached the same conclusions (and subsequently, the same modest success).

to a certain extent, RTFM really is key in life. if i've learned anything as an academic researcher, it's that we read preciously few articles carefully (if at all). the key, i suppose, is to choose those few very carefully, and do a good job on reading those selected few.

[1]
i should emphasize the "for work" part of that stipulation. probably i'll take a week or two off, here and there. already some friends and i are planning a camping trip sometime in july.

[2]
on the other hand, part of me wonders what would happen if i were able to solve those problems, all at once. i wouldn't know exactly what to work on next. i think i'd actually become slightly depressed at losing a long-standing adversary, to be honest.

[3]
when it comes to environmentalism, i have to admit that my reasons are particularly self-serving. you see, i like to hike the outdoors (despite my dislike of travel) and human-wrought desolation just pisses me off. i don't care if most humans on this world will not get the chances that i do to explore this world. i still want mine.

Wednesday, June 12, 2013

earlier in the year i had thought that this summer would be very productive. i'd set aside long blocks of time in order to get things done .. but being at the start of a 7+ week stretch, free of traveling, all i seem to do is procrastinate and occasionally daydream.

i did try and work today. in particular i tried to get rid of some stars (or asterisks) in one of my manuscripts, but realised that it would be harder than i thought. (so, as before, i am still stuck at three.)

i don't think i can do better. maybe i should just submit the damned thing.

"As the speaker is cycled through various frequencies the sand naturally gravitates to the area where the least amount of vibration occurs causing fascinating geometric patterns to emerge. There’s actually a mathematical law that determines how each shape will form, the higher the frequency the more complex the pattern."

#4: some rather old maths homework: Two math-notebook pages recently authenticated as belonging to Abraham Lincoln suggest the 16th president, who was known to downplay his formal education, may have spent more time in school than usually thought.

#5: okay, maybe a little bit of news.. but i'll restrict them to headlines and one-sentence excerpts. those headlines that are self-explanatory or sound inherently interesting (see below) won't even have an excerpt at all.

Wednesday, June 05, 2013

on a completely mathematically-unrelated subject, i was told today by a colleague that his girlfriend thought i was 23 years old [1].

upon hearing this, i spat out: "WHAT" ..?
then he pointed out: "you should be flattered" ..!

to be fair, he does have a point; then again, so do i:

what came first to mind was how agonising life was at 23;

it was early in my ph.d.,
i probably hadn't passed all my qualifying exams yet,
and definitely hadn't started working with the advisor yet.

in general i wasn't sure of anything or whether this was the right path to take.

heck, i still don't know.

the only reason i'm so relaxed about it now is that my imagination's gotten worse, and the subsequent worst-case scenarios have stopped looking so gloomy! (-:

[1]
as for some background, he's probably about 24-25 years old (though i've actually never asked him) and i surmise that his girlfriend is the same age. when i met her over dinner, there was also a masters' student joining us too, so guessing beyond the age of 30 would be a little out of context.

Monday, June 03, 2013

the discussion is technical, but it's really clear; then again, maybe i'm too used to these kind of details by now. they are all carefully present in the write-up, though. not to make too much of it, but the discussion almost has a nice flow to it.

that said, i think i'm a lot looser with my writing now and perhaps less careful .. which is a bad thing. i guess we all pick up bad habits as we get older. 7-:

// added: 6 june 2013 @ 15:09 EEST

ye gods: under the journal's formatting guidelines, the manuscript is 38 pages long!

years ago i thought it would be "cheating" to split it up into two papers. my original intention was to (A) solve a special case of a conjecture, and so doing (B) build up the basics of a theory, which might actually be useful someday. at the time, however, i doubted that each piece could stand on its own ..

.. but now i'm not so sure. the proof of that special case takes 10 pages on its own, so it would have been fine to write a short note and cite it in the other part; that way, the theory-building part of the paper would be justified (but only that it would not be in the same writeup).

[sighs]

oh well; it's too late now. after five years of waiting, i just don't care anymore about what might have beens ..

i've heard the claim before that for some phenomena, physicists "knew" them long before mathematicians did.

that choice of wording has always bothered me, because it suggests that (experimental) physicists are somehow "smarter" than mathematicians .. as if the disciplines can really be compared. it's a difference in methodology, or for that matter, epistemology:

scientific conclusions are only as accurate as the data you can physically observe and measure within the scope of the problem. on the other hand, if a mathematician proves that something is true then (s)he is simultaneously asserting that a countless number of pathologies are impossible.

at any rate .. with respect to the same abuse of language, sometimes artists are "smarter" than mathematicians:

"
The same is true in visual arts. Vincent van Gogh’s later paintings had all sorts of swirling, churning patterns in the sky — clouds and stars that he painted as if they were whirlpools of air and light. And, it turns out, that’s what they were! In 2006, physicists compared van Gogh’s patterns of turbulence with the mathematical formula for turbulence in liquids. The paintings date to the 1880s. The mathematical formula dates to the 1930s. Yet van Gogh’s turbulence in the sky provided an almost identical match for turbulence in liquid.

Art sometimes precedes scientific analysis, and the relationship can go the other way too: Scientists can use art to understand math.

Even the seemingly random splashes of paint that Jackson Pollock dripped onto his canvases show that he had an intuitive sense of patterns in nature. In the 1990s, an Australian physicist, Richard Taylor, found that the paintings followed the mathematics of fractal geometry — a series of identical patterns at different scales, like nesting Russian dolls. The paintings date from the 1940s and 1950s. Fractal geometry dates from the 1970s. That same physicist discovered that he could even tell the difference between a genuine Pollock and a forgery by examining the work for fractal patterns."

the last paragraph gives me pause: to what extent is this phenomenon unique, really? the only way to check this if most drip paintings do not obey a self-similar pattern; otherwise pollock wouldn't really be unique in this sense, would he?

"
Survivorship bias also flash-freezes your brain into a state of ignorance from which you believe success is more common than it truly is and therefore you leap to the conclusion that it also must be easier to obtain. You develop a completely inaccurate assessment of reality thanks to a prejudice that grants the tiny number of survivors the privilege of representing the much larger group to which they originally belonged."

(this excerpt is a lot more interesting, once you learn the title of the article.)

i like this article, if only because it hints at why we mathematicians study things that "don't really exist" .. at least in everyday physical reality.

"
What these theories do share is a certain level of rigor. Rather than being arbitrary, they involve precisely defined conditions that collectively give rise to interesting properties. While a theory in the theoretical physics sense isn't “true” in that it doesn’t describe the real world, it is “true” in that two researchers will agree on the theory’s properties. This allows interested parties to build off each other’s work.
"

in case you were wondering ..

yes, the names of well-known people and theorems are obscured. as for why, too many irrelevant Google searches find their way here.
for example, search: "mαth jοbs wιkι" but without the funny greek symbols. as of a year or two ago, this blog came up in the first few hits.